For dimension three or higher, the conjecture is still largely open, with only a few partial results known. Kai Tang in [1]} proved the conjecture under the additional assumption that the metric is Chern Kähler-like, meaning that the curvature tensor \(R\)  obeys all the Kähler symmetries. In their recent paper [2]}, Chen-Chen-Nie proved the conjecture for the case \(c\le 0\)  under the additional assumption that \(g\)  is locally conformally Kähler. They also pointed out the necessity of the compactness assumption in the conjecture by explicit examples.
